Targeted Maximum Likelihood Based Causal Inference

Given causal graph assumptions, intervention-specific counterfactual distributions of the data can be defined by the so called G-computation formula, which is obtained by carrying out these interventions on the likelihood of the data factorized according to the causal graph. The obtained G-computation formula represents the counterfactual distribution the data would have had if this intervention would have been enforced on the system generating the data. A causal effect of interest can now be defined as some difference between these counterfactual distributions indexed by different interventions. For example, the interventions can represent static treatment regimens or individualized treatment rules that assign treatment in response to time-dependent covariates, and the causal effects could be defined in terms of features of the mean of the treatment-regimen specific counterfactual outcome of interest as a function of the corresponding treatment regimens. Such features could be defined nonparametrically in terms of so called (nonparametric) marginal structural models for static or individualized treatment rules, whose parameters can be thought of as (smooth) summary measures of differences between the treatment regimen specific counterfactual distributions. In this article, we provide templates for implementation of the targeted maximum likelihood estimator of causal effects of multiple time point interventions. This involves the use of loss-based super-learning to obtain an initial estimate of the unknown factors of the G-computation formula, and subsequently, applying a target-parameter specific optimal fluctuation function (least favorable parametric submodel) to each estimated factor, estimating the fluctuation parameter(s) with maximum likelihood estimation, and iterating this updating step till convergence. The targeted maximum likelihood step makes the resulting estimator of the causal effect double robust in the sense that it is consistent if either the initial estimator is consistent, or the estimator of the optimal fluctuation function is consistent. The optimal fluctuation function is correctly specified if the conditional distributions of the nodes in the causal graph one intervenes upon are correctly specified. The latter conditional distributions often comprise the so called treatment and censoring mechanism. Selection among different targeted maximum likelihood estimators (e.g., indexed by different initial estimators) can be based on loss-based crossvalidation such as likelihood based cross-validation or cross-validation based on another appropriate loss function for the distribution of the data. Some specific loss functions are mentioned in this article. In this article, a variety of interesting observations about targeted maximum likelihood estimation are made, and a concrete template for the practical implementation of targeted maximum likelihood estimation is presented as well. In addition, we demonstrate it for estimation of a causal effect of dynamic treatment rules defined in terms of a marginal structural working model, inspired by HIV applications.